The ring that closes at every size

An ordinary scissor is two straight bars crossed and pinned at their middles. Chain them end to end and the assembly grows in a straight line: a lazy tong, a gate, a lifting table.

Bend each bar at its pivot and something else happens. The chain no longer grows straight; it curves, and if the bend is chosen correctly it curves by exactly the right amount to come back to where it started. A ring of eight such pairs opens and shuts like an iris, and every bar in it stays rigid.

Grübler’s count says it cannot move — and the count is the site’s own first instrument, used correctly, on a mechanism it cannot see.

The invariant #

The mechanism rests on one fact about a pair of elements, and the fact can be measured before any ring is built.

Take two identical bars, each with two arms of equal length meeting at a kink of angle $\beta$, and pin them to each other at their kinks, one the mirror image of the other. That assembly has one freedom: the relative rotation about the pivot.

It has four free ends. Two of them — one from each bar — are the pins by which the pair joins its neighbour on one side; the other two join its neighbour on the other side. Draw the line through the first two and the line through the second two. Those are the pair’s connection lines.

The angle between them is 135.000000000° at every opening — the element’s own kink angle, and nothing else. Not approximately, not for one arm length: at openings of 0.35, 0.9, 1.6 and 2.3 radians the four measurements agree to the last figure printed, and the spread across them is $10^{-15}$ radians.

Now do the same with straight bars, $\beta = \pi$.

Parallel, to $2.5 \times 10^{-14}$ radians. Which is what a lazy tong is: every connection line in a tong is parallel to every other, which is why it grows in one direction and closes on nothing.

Why the invariant is true #

The theorem is short enough to prove and short enough to get the wrong way round, and getting it the wrong way round makes it vanish, so it is worth doing.

Put the pivot at the origin. Let one element’s arms point at angles $\theta_P$ and $\theta_Q$, with $\theta_Q - \theta_P = \beta$. The other element is its mirror image, rotated by the opening $\psi$, so its arms point at $\psi - \theta_P$ and $\psi - \theta_Q$.

Now the part that has to be right: the end of the second element that lies on the first connection line is the image of the far end of the first element, not of the near one. Pairing them the other way makes both connection lines parallel at every opening and the theorem disappears entirely.

With the pairing right, the connection line through the ends at $\theta_P$ and $\psi - \theta_Q$ has direction

$$\frac{\psi - \beta}{2} + \frac{\pi}{2},$$

by the sum-to-product identity for the difference of two unit vectors, and the other line has direction $(\psi + \beta)/2 + \pi/2$. Subtracting, the angle between them is $\beta$, with the opening $\psi$ cancelling exactly.

The opening cancelling is the whole content. A quantity that survives the mechanism’s only freedom is an invariant, and an invariant of this kind is what a network needs, because it is the thing that can be made to match across every joint in an assembly at once.

Why that makes a ring #

A ring of $n$ pairs, arranged with $n$-fold symmetry, joins its pairs along radii. Consecutive radii are separated by $2\pi/n$. So the pair’s two connection lines must differ by that angle, and the invariant says they differ by $\pi - \beta$. Hence

$$\beta = \pi - \frac{2\pi}{n}.$$

Eight pairs need a kink of 135°, six need 120°, ten need 144°, twelve need 150°. A straight bar is $\beta = \pi$ and needs $n = \infty$: an ordinary scissor ring closes nowhere, which is the same sentence as the tong’s, read from the other end.

The condition, derived rather than quoted #

That argument is quick and it is worth having the closure done the long way as well, because the long way produces the deployment.

Impose $n$-fold rotational symmetry and one mirror. The two pins joining consecutive pairs then lie on the same radial line — one at radius $r_1$ and one at radius $r_2$ — and each pair’s pivot sits on the bisector between two radii, at radius $\rho$. Three unknowns.

An element runs from a pin at $(r_1, 0)$ through its kink at $(\rho, \phi/2)$ to a pin at $(r_2, \phi)$, with $\phi = 2\pi/n$. Writing its three side lengths in those unknowns and requiring the two arms to be equal gives

$$r_1 + r_2 = 2\rho\cos(\phi/2), \qquad \rho^2 - r_1 r_2 = L^2,$$

and substituting them into the third — the chord $c$ between the element’s two end pins — leaves

$$c^2 = 4L^2\cos^2(\phi/2).$$

That is a condition on the element alone, with $\rho$ nowhere in it. So $\rho$ is free, and the mechanism deploys. In terms of the kink, $c = 2L\sin(\beta/2)$ gives $\cos\beta = -\cos\phi$, which is $\beta = \pi - \phi$ again.

Both routes are in the machinery, and they agree: the ring built from the three formulae above reports a measured kink of 135.0000° for eight pairs and 150.0000° for twelve, with the arms coming out at 1.000000000 and the chord at 1.847759 against a predicted 1.8477590650225735.

What deploying actually does #

Having $\rho$ free is the mechanism; what it does to the ring is worth looking at, because it is not what the demonstrations look like they are doing.

The range has ends that nobody picked. At $\rho = L$ the discriminant of the quadratic gives $r_1 = 0$: the inner pins arrive at the centre and the ring is shut. At $\rho = L/\sin(\phi/2)$ the two radii meet and every element lies along a radius; past that there is no configuration at all, and asking for one is refused.

Between them the outer circle grows by a factor of 1.414 — a little over forty per cent — while the inner one goes from nothing to 2.400 in the same units. So a ring of this kind is very much less a thing that gets bigger than a thing that closes its aperture. That is what the mechanism is for. It is not the impression a photograph of one gives.

The count, and what it misses #

Sixteen bodies and twenty-four pins is $3 \times 15 - 2 \times 24 = -3$ grounded, or nought free. Either way: a structure.

The rank of the constraint Jacobian is 44 of 48 at eight pairs, 26 of 30 at five, 68 of 72 at twelve. Four freedoms every time — three rigid motions and the deployment — and four dependencies every time.

Those four are the count’s error and they are not an accident. They are what choosing the kink angle bought: with $\beta = \pi - \phi$, one condition per pair going round the ring is implied by the others, and four such implications survive as independent statements — dependencies among the constraints in the exact sense the field uses. Change the kink by a degree and they vanish, the rank rises to 48, the count becomes right, and the mechanism becomes a structure that cannot be assembled at the drawn radius.

The dependencies are the design, and this is the clearest example of that sentence in the field.

The check that earns it #

A closed form that returns a plausible answer for every input is a hazard this site has met before, most recently in a strand routine that produced a smooth, closed, entirely wrong path. So the ring is checked on quantities the construction does not contain.

Every element’s arms must be the same length at every deployment, because the element is rigid. Across the whole range, sampled forty times, the spread in arm length is $1.2 \times 10^{-15}$. Every element’s chord must be the same, and it is, to the same order. And the measured kink angle must equal $\pi - 2\pi/n$, which it does to nine figures at every size.

Those are the quantities a wrongly built ring would break. The radii and the pivot radius come out of the closed form and cannot disagree with it; the arms and the chord are read off the drawn positions and can.

Reading the ring against the tong #

The two mechanisms are worth putting side by side, because between them they cover the field’s whole range and they differ in one number.

A tong of five units: ten bodies, thirteen pins, four independent loops, count four, measured four, no dependencies. A ring of eight pairs: sixteen bodies, twenty-four pins, nine loops, count nought, measured four, four dependencies. Both are chains of a repeated two-body unit pinned in the same way; the only difference between the units is whether the bar is straight.

That difference does nothing to the graph, so it does nothing to the count. It does everything to the rank, because it decides whether going round the assembly the constraints come back to a statement already made.

There is a design reading of that too. A tong is easy to make: its count is right, so nothing about its behaviour depends on its dimensions being exact, and a unit cut a degree out gives a tong that reaches slightly less far. A ring is not: its behaviour depends on a hundredth of a degree in one angle, and a ring whose elements are a degree out does not open badly — it does not go together at all.

What this shares with the rest of the field #

Three things, and they are the field’s argument in miniature.

The unit’s property is not the assembly’s. A single pair of angulated elements has one freedom and no dependencies at all; it is an ordinary mechanism and its count is right. Eight of them in a ring have four freedoms and four dependencies. Nothing about the pair predicts that, and the condition that produces it — the relation between the kink and the pair count — is a property of the tiling.

The condition is one condition, repeated. A ring of eight has four dependencies and needs one number to be right. That is the same economy a Miura sheet gets from congruent vertices: build the assembly from copies of one unit and the hundred conditions collapse to one.

And the count is a statement about a difference. Nought is a true report of $m - s$ on this mechanism. It is a useless report of $m$.

One number, chosen once #

There is a way of describing what the kink angle does that generalises past this mechanism, and it is the reason the ring earns a rung of its own rather than a paragraph in the ledger’s essay.

An assembly of $N$ repeated units has, in general, a number of compatibility conditions that grows with $N$ — on a folded sheet it is $(n-2)^2$, a hundred at a hundred and forty-four panels. Solving them one at a time is not a design method. What makes a deployable buildable is finding a single parameter of the unit such that all of the conditions become the same condition, and then choosing it.

Here that parameter is the kink and the condition is $\beta = \pi - 2\pi/n$. On a Miura sheet it is the sector angle and the congruence of the vertices. In both cases the assembly’s difficulty has been moved off the assembly and onto the unit, where there is one number to get right instead of a hundred.

It is also why this field’s mechanisms all look like tessellations, and why a deployable made of units that are all slightly different is a research problem rather than a product. The alternative to symmetry is solving a hundred equations in two hundred and forty-two unknowns with no structure to exploit.

The ring at other sizes #

Running the same construction at five sizes shows what depends on the pair count and what does not.

The kink runs 108°, 120°, 135°, 144°, 150° for five, six, eight, ten and twelve pairs — always $180° - 360°/n$, so a larger ring is made of straighter elements and a ring of very many pairs is very nearly a lazy tong bent round.

The chord between an element’s two end pins runs 1.618034, 1.732051, 1.847759, 1.902113 and 1.931852 on unit arms, climbing towards 2 as the elements straighten. Those are $2\cos(\pi/n)$, and the first of them is the golden ratio for the reason that a regular pentagon’s diagonal is.

And the deployment range widens with the count: the upper limit is $L/\sin(\pi/n)$, which is 1.701 at five pairs and 3.864 at twelve. A larger ring travels further in absolute terms and does not open by a larger factor.

What does not change is the pair of nullities. Four freedoms and four dependencies at five pairs and at twelve, with the count at nought throughout. That is worth stating because it is the opposite of the folded sheet’s behaviour, where the dependency count grows as the square of the size. Here the assembly is a single cycle rather than a patch, so there is one closure to be satisfied however many pairs are in it, and the redundancy does not accumulate.

The two together bracket what a repeated unit can do to an arithmetic: a chain of them where nothing accumulates and the count is right, a cycle of them where a fixed amount is repaid once, and a patch of them where the error grows with the area.

The kink is the polygon’s own corner #

The condition $\beta = \pi - 2\pi/n$ is derived twice above and it has a reading that makes it memorable rather than merely correct, and the reading is exact.

$\pi - 2\pi/n$ is the interior angle of a regular $n$-gon. So the kink angles the table reports are not a sequence of awkward numbers: 108° is a pentagon’s corner, 120° a hexagon’s, 135° an octagon’s, 144° a decagon’s, 150° a dodecagon’s. Every one of them is an angle anybody can look up or construct.

An angulated element is bent to the corner of the polygon its ring is going to approximate. That is the whole design rule, it needs no arithmetic, and it can be read off a drawing of the target shape: draw the polygon, measure a corner, bend the element to it.

Which also says why the ring closes at every size, in a form the algebra makes true and does not make obvious. A ring of $n$ pairs is a polygon of $n$ sides whose corners are the elements’ kinks, and an element bent to that polygon’s corner fits it — at every scale, because the polygon’s angles do not depend on its size. The deployment slides the polygon’s vertices out along its own radii, which changes the side length and leaves every angle where it was.

The straight element falls out of the same expression as the limiting case, and it is worth checking it does. Set $\beta = \pi$ and solve: $\pi = \pi - 2\pi/n$ gives $2\pi/n = 0$, so $n$ is infinite. A straight-barred scissor chain is the angulated ring of infinitely many pairs — which is to say a straight line, the polygon with no corners — and that is exactly the mechanism the lazy tong is. The tong and the ring are the two ends of one family, and the kink angle is the parameter between them.

That gives the family a shape the two mechanisms did not obviously have. A tong grows along a line and never closes; a ring of twelve closes with a nearly straight element bent by thirty degrees; a ring of five closes with a sharply bent one. The more the element is bent, the tighter the ring it closes — and the relation between the two is the polygon, exactly, with no approximation anywhere in it.

What is not here #

Force, as everywhere in this field and on this site. A deployable ring’s real design problem includes what holds it open and what it costs to move; none of that is computed, and every number above survives with every force in the mechanism unknown.

Thickness. Every element here is a line, and a real angulated element is a plate with a hole in each end. Giving the members thickness is a genuine and difficult problem — the pins stop being coincident and the mechanism stops being the one analysed — and it is not addressed by scaling anything.

And the choice of $n$. The arithmetic says which kink a given pair count needs; it says nothing about which pair count to want. That is decided by how far the aperture has to open and by what the elements can be made from, and both of those are outside a field whose every quantity is a rank or a length.